Two novel polyphosphoesters containing anthracene- and furan-derived aminophosphonate moieties, namely, poly[oxyethylene(aminophosphonate-co-H-phosphonate)]s P-12 and P-13, were synthesized through an addition of poly(oxyethylene H-phosphonate) to 9-anthrylidene-furfurylamine and characterized. The novel polyphosphoester P-12 and a series of previously described anthracene-derived compounds including Schiff bases S-1 and S-2, α-aminophosphonates A-3–A-6, bis-aminophosphonate B-6, two enantiomers A-5a and A-5b, and polyphosphoesters P-8–P-11 containing aminophosphonate units were screened for antitumor activity against a panel of human leukemic cell lines, using cisplatin as a reference cytotoxic agent. As concluded from the cytotoxicity assays, both precursors S-1 and S-2 presented similar cytotoxicity profiles that are cisplatin-like only in the REH cell line. Leader compound of the α-aminophosphonates is A-4 with cell death-inducing properties fully equaling those of the referent drug in all of the screened leukemic cell lines with the only exception being the AML histological subtype HL-60. Some of the polymeric analogues elicited moderate (P-10 and P-12) to low (P-11) cytotoxic activity, whereas the polyphosphoesters P-8 and P-9 produced in vitro antitumor effects largely surpassing cisplatin’s. The compounds P-8, P-9, and A-4 could be potential new materials for anticancer therapeutic purposed.